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I am wondering if it is possible to derive a result on the stationarity of the positive or negative part of a covariance stationary process.

Namely, consider $\{ X_t \}, t=1,2,3,...,$ a covariance stationary time series, i.e. with two first moments constant in time: for all $t$, $\mathbb{E}[X_t] = x$ and $\text{var}(X_t) = y$. My question is the following: are $X_t^+ = \max(X_t, 0)$ and $X_t^- = \min(X_t, 0)$ covariance stationary? I did not manage to derive a proof or to find a counter exemple. Any ideas? Many thanks.

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